Just how trump cosplays as a working class sympathiser, so did the nazis. They tapped into the popular sentiments at the time (socialism) and redirected them into blaming scapegoats (commies, disabled people, jews, gender and sexual minorities). It probably sounds quite familiar.

i am pretty sure my understanding of this topic comes from Ishay Landa’s The Apprentice’s Sorcerer, a compelling read.

anyway…

Did they call themselves socialist just as branding?

yes. in the well-cited book above, Landa talks about the dangers of “taking fascists at their word” for their beliefs. the color red and use of the word “socialism” were specifically chosen and meant to disrupt and confound the emerging socialist movement by creating uncertainty and confusion among the frustrated, alienated working class gravitating towards mass action.

the fascists claimed to be socialist to angry workers in public, but were much more circumspect in their language when courting the elites and their capital formations, which they did extensively. these relationships and the fascists willingness to destroy trade unionists and suppress strikes made them the darlings of capital and they found eager partners in supplying slave labor to the capitalists.

there were probably some among the nazis that believed the movement was meant to elevate the common worker, but after the Night of the Long Knives the writing was on the wall and there would be no organizing among whatever was left of that faction after most vocal proponents had been liquidated.

by the time the nazis began their invasion, they enjoyed lots of support from aristocrats and business leaders all over the world and all the socialists were joining resistance groups, dead or in camps.

have lost the provenance for this but the quotes check out

The Nazis were not socialists. Their entire goal was to latch onto a popular political movement and redefine it to fit their needs(as all fascists typically do).

They did not support worker ownership of the means of production and the right for workers to work for themselves. Hitler repealed legislation that nationalized industry in Germany, and oversaw the expansion of private industry. The first modern implementation of privatization on a grand scale took place under the supervision of the Nazis. The word “privatization” was coined to describe a central tenet of Nazi economic policy. The Nazis raided and imprisoned union leaders and broke up trade unions. They repealed worker rights.

Behold Hitler’s own words:

“There are only two possibilities in Germany; do not imagine that the people will forever go with the middle party, the party of compromises; one day it will turn to those who have most consistently foretold the coming ruin and have sought to dissociate themselves from it. And that party is either the Left: and then God help us! for it will lead us to complete destruction - to Bolshevism, or else it is a party of the Right which at the last, when the people is in utter despair, when it has lost all its spirit and has no longer any faith in anything, is determined for its part ruthlessly to seize the reins of power - that is the beginning of resistance of which I spoke a few minutes ago.”

—Hitler, explaining that he vehemently opposes the Left, and believes only Rightists like himself can make Germany great again.

“Our adopted term ‘Socialist’ has nothing to do with Marxian Socialism. Marxism is anti-property; true socialism is not.”

—Hitler, literally admitting his “socialism” is a whole new thing and has nothing to do with the usual definition of the word.

“The ideology that dominates us is in diametrical contradiction to that of Soviet Russia. National Socialism is a doctrine that has reference exclusively to the German people. Bolshevism lays stress on international mission. We National Socialists believe a man can, in the long run, be happy only among his own people.”

—Hitler, trying so hard to explain that he isn’t a socialist, that he opposes socialism, and that the term National Socialist is something he made up and only has meaning within the context of its own paradigm.

“We National Socialists see in private property a higher level of human economic development that according to the differences in performance controls the management of what has been accomplished enabling and guaranteeing the advantage of a higher standard of living for everyone. Bolshevism destroys not only private property but also private initiative and the readiness to shoulder responsibility.”

—Hitler, spelling it out in very clear terms that he wholeheartedly supports private ownership of property, i.e. capitalism, and opposes worker ownership of property, which he calls “Bolshevism”, i.e. real, actual socialism.

“What right do these people have to demand a share of property or even in administration?.. The employer who accepts the responsibility for production also gives the workpeople their means of livelihood. Our greatest industrialists are not concerned with the acquisition of wealth or with good living, but, above all else, with responsibility and power. They have worked their way to the top by their own abilities, and this proof of their capacity – a capacity only displayed by a higher race – gives them the right to lead.”

—Hitler, attacking the notion of worker ownership of property and licking capitalist boot.

They were stealing leftist valor.